Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x⁶-9x⁴-16x²+144

Verified step by step guidance

1. Start by recognizing that the polynomial ƒ(x) = x^6 - 9x^4 - 16x^2 + 144 is a polynomial in terms of powers of x^2. To simplify, let’s use a substitution: let \( y = x^2 \). This transforms the polynomial into a cubic in y: \( y^3 - 9y^2 - 16y + 144 = 0 \).
2. Next, solve the cubic equation \( y^3 - 9y^2 - 16y + 144 = 0 \) for y. You can try to find rational roots using the Rational Root Theorem by testing possible factors of the constant term (144) over factors of the leading coefficient (1).
3. Once you find a root \( y = a \), perform polynomial division or synthetic division to factor the cubic into \( (y - a)(quadratic) = 0 \). Then solve the quadratic factor using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the other values of y.
4. After finding all values of y, recall that \( y = x^2 \). For each value of y, solve for x by taking the square root: \( x = \pm \sqrt{y} \). This will give you the complex zeros of the original polynomial.
5. List all zeros, including real and complex ones, making sure to include both the positive and negative square roots for each y-value. These are the exact complex zeros of the polynomial function.

Key Concepts

Complex Zeros of Polynomial Functions

Complex zeros are the values of x, including real and non-real numbers, that make the polynomial equal to zero. Finding all complex zeros involves solving the polynomial equation, often requiring factoring or applying theorems to identify roots beyond just real numbers.

Factoring Higher-Degree Polynomials

Factoring polynomials of degree higher than two often involves recognizing patterns such as quadratic forms or using substitution to reduce the polynomial to simpler factors. This step is crucial to break down the polynomial into products of lower-degree polynomials, making it easier to find zeros.

Use of Substitution Method

The substitution method involves replacing a complex expression (like x^2) with a single variable to simplify the polynomial into a quadratic or another solvable form. After solving for the new variable, substitute back to find the original variable's values, including complex roots.